What is the branch of mathematics called "Category Theory" all about? From the introduction, titled "Please read this", of Conceptual Mathematics: A First Introduction to Categories by F William Lawvere and Stephen Hoel Schanuel:

We all begin gathering mathematical ideas in early childhood, when we discover that our two hands match, and later when we learn that other children also have grandmothers, so that this is an abstract relationship that a child might bear to an older person, and then that 'uncle' and 'cousin' are of this type also; when we tire of losing at tic-tac-toe and work it all out, never to lose again; when we first try to decide why things look bigger as they get nearer, or whether there is an end to counting.

As the reader goes through it, this book may add some treasures to the collection, but that is not its goal. Rather we hope to show how to put the vast storehouse in order, and to find the appropriate tool when it is needed, so that the new ideas and methods collected and developed as one goes through life can find their appropriate places as well. ...